Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Predictions deviated systematically from algebra students' performances but closely matched a view implicit in textbooks. Analysis of students' problem-solving strategies indicates specific ways that students' algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students' performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.
Mitchell J. Nathan, Assistant Professor, School of Education, University of Colorado, Campus Box 249, Boulder, CO 80304; firstname.lastname@example.org
Kenneth R. Koedinger, Senior Research Scientist, Human-Computer Interaction Institute, Carnegie Mellon University, Pittsburgh, PA 15213; Koedinger@cmu.edu